Undim family: Difference between revisions

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== Undim ==

Undim features a quarter-octave period, which acts as the [[1215/1024|ptolemaic augmented second (1215/1024)]]. That and five [[4/3|perfect fourths]] (i.e. a minor second, ~[[256/243]]) give the interval class of 5. Undim equates the [[Pythagorean comma]] with a stack of four [[schisma]]s~~. This makes~~ it a member of the [[schismic–Pythagorean equivalence continuum]], with {{nowrap| ''n'' {{=}} 4 }}.

Undim features a quarter-octave period, which acts as the [[1215/1024|ptolemaic augmented second (1215/1024)]]. That and five [[4/3|perfect fourths]] (i.e. a minor second, [[~]][[256/243]]) give the interval class of 5. Undim has a [[ploidacot]] of tetraploid monocot, and equates the [[Pythagorean comma]] with a stack of four [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]] with {{nowrap| ''n'' {{=}} 4 }}.

Note that all versions of undim (ones that do not already map 19 differently and more accurately) have an obvious extension to prime 19 by observing that sharpening 1215/1024 by [[1216/1215]] results in [[19/16]], thus mapping 19/16 to [[4edo|1\4]]. This interpretation is arguably much more harmonically plausible, owing to its simplicity and thereby greater tolerance to mistuning.

The name ''undim'' was given by [[Petr Pařízek]] in 2011 for it is some sort of opposite to [[diminished (temperament)|diminished]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.

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=== Overview to extensions ===

The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at. Septimal undim ({{nowrap| 140 & 152 }}) tempers out 5120/5103 (hemifamity). Unlit ({{nowrap| 152 & 316 }}) does 4375/4374 (ragisma) instead. Twilight ({{nowrap| 152 & 176 }}) adds 6144/6125 (porwell) to the comma list and splits the period into two – 1/8 of an octave.

Note that all versions of undim (ones that do not already map 19 differently and more accurately) have an obvious place for prime 19 by observing that sharpening 1215/1024 by [[1216/1215]] results in [[19/16]], thus mapping 19/16 to [[4edo|1\4]].

== Septimal undim ==

Septimal undim tempers out the [[dimcomp comma]], mapping ~25/21 to the 1/4-octave period. It can be described as {{nowrap| 12 & 140 }}, and is the unique temperament that equates a syntonic~septimal comma with a stack of three [[marvel comma]]s. A [[Pythagorean comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma (interval region)|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[292edo]] makes for an excellent tuning.

Septimal undim tempers out the [[dimcomp comma]], mapping [[~]][[25/21]] to the 1/4-octave period. It can be described as {{nowrap| 12 & 140 }}, and is the unique temperament that equates a syntonic~septimal comma with a stack of three [[marvel comma]]s. A [[Pythagorean comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma (interval region)|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[292edo]] makes for an excellent tuning.

[[Subgroup]]: 2.3.5.7

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== Unlit ==

Unlit tempers out 4375/4374, the [[ragisma]], and may be described as {{nowrap| 152 & 164 }}. This temperament is more faithful to the 5-limit optimal tuning, but comes with the cost of a much higher complexity.

[[Subgroup]]: 2.3.5.7

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== Twilight ==

Twilight tempers out 6144/6125, the [[porwell comma]], and may be described as {{nowrap| 152 & 176 }}. It splits the period of undim into 1/8 of an octave, and has a [[ploidacot]] of octaploid monocot.

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~7168/6561, ~3

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 == Undim ==
-Undim features a quarter-octave period, which acts as the [[1215/1024|ptolemaic augmented second (1215/1024)]]. That and five [[4/3|perfect fourths]] (i.e. a minor second, ~[[256/243]]) give the interval class of 5. Undim equates the [[Pythagorean comma]] with a stack of four [[schisma]]s. This makes it a member of the [[schismic–Pythagorean equivalence continuum]], with {{nowrap| ''n'' {{=}} 4 }}.
+Undim features a quarter-octave period, which acts as the [[1215/1024|ptolemaic augmented second (1215/1024)]]. That and five [[4/3|perfect fourths]] (i.e. a minor second, [[~]][[256/243]]) give the interval class of 5. Undim has a [[ploidacot]] of tetraploid monocot, and equates the [[Pythagorean comma]] with a stack of four [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]] with {{nowrap| ''n'' {{=}} 4 }}.
-Note that all versions of undim (ones that do not already map 19 differently and more accurately) have an obvious extension to prime 19 by observing that sharpening 1215/1024 by [[1216/1215]] results in [[19/16]], thus mapping 19/16 to [[4edo|1\4]]. This interpretation is arguably much more harmonically plausible, owing to its simplicity and thereby greater tolerance to mistuning.
 The name ''undim'' was given by [[Petr Pařízek]] in 2011 for it is some sort of opposite to [[diminished (temperament)|diminished]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.
@@ Line 28: / Line 26: @@
 === Overview to extensions ===
 The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at. Septimal undim ({{nowrap| 140 & 152 }}) tempers out 5120/5103 (hemifamity). Unlit ({{nowrap| 152 & 316 }}) does 4375/4374 (ragisma) instead. Twilight ({{nowrap| 152 & 176 }}) adds 6144/6125 (porwell) to the comma list and splits the period into two – 1/8 of an octave.
+Note that all versions of undim (ones that do not already map 19 differently and more accurately) have an obvious place for prime 19 by observing that sharpening 1215/1024 by [[1216/1215]] results in [[19/16]], thus mapping 19/16 to [[4edo|1\4]].
 == Septimal undim ==
-Septimal undim tempers out the [[dimcomp comma]], mapping ~25/21 to the 1/4-octave period. It can be described as {{nowrap| 12 & 140 }}, and is the unique temperament that equates a syntonic~septimal comma with a stack of three [[marvel comma]]s. A [[Pythagorean comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma (interval region)|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[292edo]] makes for an excellent tuning.
+Septimal undim tempers out the [[dimcomp comma]], mapping [[~]][[25/21]] to the 1/4-octave period. It can be described as {{nowrap| 12 & 140 }}, and is the unique temperament that equates a syntonic~septimal comma with a stack of three [[marvel comma]]s. A [[Pythagorean comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma (interval region)|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[292edo]] makes for an excellent tuning.
 [[Subgroup]]: 2.3.5.7
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 == Unlit ==
+Unlit tempers out 4375/4374, the [[ragisma]], and may be described as {{nowrap| 152 & 164 }}. This temperament is more faithful to the 5-limit optimal tuning, but comes with the cost of a much higher complexity.
 [[Subgroup]]: 2.3.5.7
@@ Line 126: / Line 128: @@
 == Twilight ==
+Twilight tempers out 6144/6125, the [[porwell comma]], and may be described as {{nowrap| 152 & 176 }}. It splits the period of undim into 1/8 of an octave, and has a [[ploidacot]] of octaploid monocot.
 [[Subgroup]]: 2.3.5.7
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 {{Mapping|legend=1| 8 0 82 -79 | 0 1 -5 8 }}
 : mapping generators: ~7168/6561, ~3